The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let \(S_{n}^{l}\) be the graph obtained by identifying the center of the star S n-l+1 with any vertex of C l . By \(R_{n}^{l,\,k}\) we denote the graph obtained by identifying one pendent vertex of the path P n-l-k+1 with one pendent vertex of \(S_{l+k}^{l}\) . In this paper, we show that \(R_{n}^{l,\,k}\) is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).
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References
Hosoya H. (1971). Bull. Chem. Soc. Jpn. 44: 2332
Gutman I. (1993). J. Math. Chem. 12: 197–210
Zhang L.Z. (1998). J. Math. Study 31: 437–441
Zhang L.Z., et al. (2001). Sci. Chn. (Series A) 44: 1089–1097
Zhang L.Z., et al. (2003). J. Math. Chem. 34: 111–122
Gutman I., et al. (1986). Mathematical Concepts in Organic Chemistry. Sringer, Berlin
Hou Y.P. (2002). Discrete Appl. Math. 119: 251–257
Yu A.M., et al. (2006). MATCH Commun. Math. Comput. Chem. 55(1): 103–118
Yu A.M., et al. (2007). Accepted by J. Math. Chem. 41: 33–43
J.P. Ou, On extremal unicyclic molecular graphs with maximal Hosoya index, Submitted.
J.P. Ou, J. Math. Chem. (2006) in press.
Bondy J.A., Murty U.S.R. (1976). Graph Theory with Applications. North-Holland, Amsterdam
Minc H. (1978). Permanents. Addison-Wesley, Reading, MA
Hua H.B. (2007). MATCH Commun. Math. Comput. Chem. 57: 351–361
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Hua, H. Hosoya index of unicyclic graphs with prescribed pendent vertices. J Math Chem 43, 831–844 (2008). https://doi.org/10.1007/s10910-007-9232-z
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DOI: https://doi.org/10.1007/s10910-007-9232-z